Inference for Dispersion and Curvature of Random Objects
针对测地度量空间中随机对象的离散性度量，推导了Fréchet方差与度量方差联合分布的中心极限定理，发现Alexandrov曲率决定两者关系，并据此提出推断空间曲率的新检验方法，适用于分布数据和点云数据。
There are many open questions pertaining to the statistical analysis of random objects, which are increasingly encountered. A major challenge is the absence of linear operations in such spaces. A basic statistical task is to quantify statistical dispersion or spread. For two measures of dispersion for data objects in geodesic metric spaces, Fréchet variance and metric variance, we derive a central limit theorem (CLT) for their joint distribution. This analysis reveals that the Alexandrov curvature of the geodesic space determines the relationship between these two dispersion measures. This suggests a novel test for inferring the curvature of a space based on the asymptotic distribution of the dispersion measures. We demonstrate how this test can be employed to detect the intrinsic curvature of an unknown underlying space, which emerges as a joint property of the space and the underlying probability measure that generates the random objects. We investigate the asymptotic properties of the test and its finite-sample behavior for various data types, including distributional data and point cloud data. We illustrate the proposed inference for intrinsic curvature of random objects using gait synchronization data represented as symmetric positive definite matrices and energy compositional data on the sphere. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.